Modelling droplet-particle interactions on solid surfaces by coupling the lattice Boltzmann and discrete element methods

Imagine a world where tiny, invisible forces are constantly playing tug-of-war on the surface of everything around us. This is the realm of multiphase flows, a branch of physics that studies how different materials—like liquids, gases, and solid bits—interact when they move together. Think of a raindrop rolling off a leaf, or dust being swept away by a breeze. In these scenarios, three main characters are always at play: hydrodynamic forces (the push and pull of the moving fluid, like water dragging a leaf), capillary forces (the sticky "skin" tension of the liquid that tries to hold things together), and friction (the resistance you feel when sliding a book across a table). Scientists have long wanted to understand exactly how these forces team up or fight each other, especially when trying to figure out how nature cleans itself or how we can design better surfaces. But in the real world, it's incredibly hard to test these forces one by one because changing the stickiness of a surface often changes how rough it is, too. It's like trying to taste just the salt in a soup without changing the temperature.

This is where a team of researchers from the University of Edinburgh steps in with a clever digital solution. They have built a sophisticated computer simulation that acts like a microscopic movie studio, allowing them to watch how a single drop of water interacts with a tiny, dusty particle on a solid surface. By combining two powerful mathematical tools—one that simulates the flow of fluids and another that simulates how solid objects bounce and roll—they created a virtual laboratory. Here, they can tweak the "stickiness" of the water, the "roughness" of the particle, and the speed of the drop independently, something that is nearly impossible to do perfectly in a physical experiment. Their goal? To finally crack the code on how drops clean surfaces, a phenomenon famously seen in the self-cleaning lotus leaf, and to see if they can predict exactly when a drop will successfully sweep away a contaminant or just roll right past it.

The Digital Dance of Drops and Dust

The researchers developed a new 3D computer method that couples the Lattice Boltzmann Method (LBM) with the Discrete Element Method (DEM). To understand what this means, imagine LBM as a grid of tiny, invisible billiard balls that bounce around to simulate how water or air flows. It's great for seeing how fluids swirl and stretch. On the other hand, DEM is like a physics engine for solid objects; it calculates how particles bounce, slide, and roll when they hit each other or a wall. The magic of this paper is that the authors figured out how to make these two systems talk to each other in three dimensions, accounting for the fact that particles can roll, slide, and even get stuck in the "skin" of a liquid drop.

Before they could trust their virtual world, they had to prove it worked. They ran a series of "stress tests" to see if their simulation matched real-world physics. First, they watched a particle bounce off a wall to make sure their model of friction and normal forces (the push-back when things touch) was accurate. They found that when they added "damping" (energy loss), the particle bounced lower each time, just like a real ball losing energy. Next, they simulated a particle floating in a flowing pipe to check their hydrodynamic forces. The particle moved to a specific spot in the pipe, a phenomenon known as the Segrè-Silberberg effect, and their simulation matched experimental data perfectly. Finally, they tested capillary forces by pulling a particle out of a liquid interface. They measured the force needed to detach the particle and found it matched mathematical predictions for a wide range of contact angles, proving their model could handle the sticky "skin" of the liquid correctly.

The Great Drop Heist: Push-Pull vs. Enter-Exit

With their tools calibrated, the team turned to the main event: watching a drop try to clean a surface. They set up a scenario where a drop slides across a surface toward a single particle. Depending on how "frictional" the particle is, two very different stories unfold.

In the "Push-Pull" scenario, the particle is slippery (low friction). As the drop hits the particle, it doesn't just push it forward. Instead, the particle gets caught on the side of the drop and starts to roll around the drop's edge, like a passenger clinging to the side of a moving bus. Eventually, the particle ends up stuck to the back of the drop, being pulled along. The simulation showed that even if the particle is perfectly aligned with the drop, tiny imperfections or "roughness" on the surface can cause it to drift to the side and start this circular journey. Interestingly, the drop actually pulls the particle down slightly as it drags it, increasing the friction and making the job harder.

In the "Enter-Exit" scenario, the particle is sticky or heavy (high friction). Here, the drop can't get the particle to roll around the side. Instead, the drop pushes the particle right into its center. The particle gets swallowed by the drop, travels across the bottom, and then pops out the other side. It's like a person walking through a revolving door: they enter, cross the middle, and exit. In this case, the drop fails to clean the surface effectively because the particle is only moving while it's touching the liquid-air interface at the very front and back of the drop. For most of the time, the particle is just sitting inside the drop, not being transported.

The Goldilocks Zone of Drop Size

One of the most fascinating findings is that drop size matters a lot. The researchers simulated drops of different sizes and found a "Goldilocks" zone for cleaning.

• Too small: A tiny drop doesn't have enough force to overcome the friction holding the particle down. It just pushes the particle a tiny bit and stops.
• Too large: A huge drop moves so fast and is so big that the particle gets swallowed (the Enter-Exit scenario) and left behind.
• Just right: A medium-sized drop moves at a speed where it can push the particle effectively without swallowing it, or it can pull it along the side efficiently.

The team also discovered that the type of friction matters more than you might think. They found that sliding friction (the resistance to sliding) is the main villain in preventing a drop from cleaning a surface. Even if the rolling friction is low, if the sliding friction is high, the particle won't move. However, if the sliding friction is low, the particle is much more likely to be captured and removed, even if the rolling friction is high. This suggests that to design a self-cleaning surface, you need to focus on making it hard for particles to slide, rather than just making them hard to roll.

Why This Matters

This work is a big step forward because it allows scientists to "tune" variables that are impossible to separate in real life. In a real experiment, if you change the chemical makeup of a surface to make it more water-repellent, you might accidentally change how rough it is, messing up your results. But in this simulation, the researchers could change the contact angle (how water beads up) without changing the friction, or change the drop speed without changing the liquid's stickiness.

The authors suggest that this method opens the door to studying many other messy, real-world problems. They mention that their approach could help us understand how rain causes soil erosion, how microplastics are transported by raindrops in the environment, and even how respiratory droplets might interact with surfaces. While they haven't solved these problems yet, they have built a powerful new microscope for the digital world that lets us watch the invisible forces of nature play out in 3D, one drop at a time. The paper concludes that while their method is a powerful tool, future work will need to expand these models to handle non-spherical particles and even rougher, more complex surfaces to truly mimic the chaotic beauty of the natural world.